A standard deck of cards contains 52 cards. One card is selected from the deck.
(a) Compute the probability of randomly selecting an ace or four.
(b) Compute the probability of randomly selecting an ace or four or three.
(c) Compute the probability of randomly selecting an eight or club.
(a) [tex]P(ace \text{ or } four) =[/tex]
(b) [tex]P(ace \text{ or } four \text{ or } three) =[/tex]
(c) [tex]P(eight \text{ or } club) =[/tex]
Answer (2)
The probabilities calculated are: (a) 0.154 for selecting an ace or four, (b) 0.231 for selecting an ace or four or three, and (c) 0.308 for selecting an eight or club.
;
Calculate the probability of selecting an ace or four: 52 4 + 52 4 = 52 8 ≈ 0.154 .
Calculate the probability of selecting an ace or four or three: 52 4 + 52 4 + 52 4 = 52 12 ≈ 0.231 .
Calculate the probability of selecting an eight or club: 52 4 + 52 13 − 52 1 = 52 16 ≈ 0.308 .
The probabilities are: (a) 0.154, (b) 0.231, (c) 0.308. 0.154 , 0.231 , 0.308
Explanation
Understanding the Problem Let's break down this probability problem step by step. We're dealing with a standard deck of 52 cards, and we want to find the probabilities of drawing specific combinations of cards.
Calculating P(Ace or Four) (a) We want to find the probability of drawing an ace or a four. There are 4 aces and 4 fours in a standard deck. Since these are mutually exclusive events (a card cannot be both an ace and a four simultaneously), we can simply add their individual probabilities.
The probability of drawing an ace is 52 4 , and the probability of drawing a four is 52 4 . Therefore, the probability of drawing an ace or a four is: 52 4 + 52 4 = 52 8 = 13 2 ≈ 0.154 (Rounded to three decimal places.)
Calculating P(Ace or Four or Three) (b) Now, let's find the probability of drawing an ace, a four, or a three. Similar to part (a), these are mutually exclusive events. There are 4 aces, 4 fours, and 4 threes in the deck.
The probability of drawing an ace is 52 4 , the probability of drawing a four is 52 4 , and the probability of drawing a three is 52 4 . So, the probability of drawing an ace, a four, or a three is: 52 4 + 52 4 + 52 4 = 52 12 = 13 3 ≈ 0.231 (Rounded to three decimal places.)
Calculating P(Eight or Club) (c) This time, we want to find the probability of drawing an eight or a club. There are 4 eights and 13 clubs in the deck. However, we need to be careful because one of the eights is also a club (the eight of clubs). If we simply add the probabilities, we'll be double-counting the eight of clubs. To avoid this, we use the following formula:
P(eight or club) = P(eight) + P(club) - P(eight and club)
The probability of drawing an eight is 52 4 , the probability of drawing a club is 52 13 , and the probability of drawing the eight of clubs is 52 1 . Therefore, the probability of drawing an eight or a club is: 52 4 + 52 13 − 52 1 = 52 16 = 13 4 ≈ 0.308 (Rounded to three decimal places.)
Final Answers In summary: (a) The probability of selecting an ace or four is approximately 0.154. (b) The probability of selecting an ace or four or three is approximately 0.231. (c) The probability of selecting an eight or club is approximately 0.308.
Examples
Understanding probability is crucial in many real-life scenarios. For instance, when playing card games like poker, knowing the probability of drawing certain cards helps you make informed decisions about betting and strategy. Similarly, in insurance, companies use probability to assess the risk of insuring individuals or assets, determining premiums based on the likelihood of certain events occurring. Even in weather forecasting, meteorologists use probability to predict the chance of rain or other weather conditions, helping people plan their activities accordingly. These examples highlight how probability plays a vital role in decision-making across various fields.
